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%%文档的题目、作者与日期
%\author{王立庆（2020级数学与应用数学1班） }
\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{常微分方程期中练习}
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\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
\date{2023 年 10 月 31 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %第1题
\begin{enumerate}
\item  求出平面上所有直线所满足的微分方程。
\item  求出平面上所有过原点的抛物线所满足的微分方程。
\end{enumerate}

\vspace{0.2cm}



\vspace{8cm}

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\item  %第2题
作出微分方程 $\frac{dy}{dx}=xy-1$ 在第一象限的线素场的草图。取下述格点。
\begin{table}[ht!]\centering
\begin{tabular}{|M{1cm}|M{1cm}|M{1cm}|M{1cm}|}\hline
$dy/dx$ & $x=0$ & $x=1$ & $x=2$ \\ \hline
$y=0$ &&& \\ \hline 
$y=1$ &&& \\ \hline 
$y=2$ &&& \\ \hline 
\end{tabular}
\end{table}

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\newpage
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\item  %第3题
判断下述方程是否为恰当方程，如果是恰当方程请求解。
\begin{eqnarray*}
(4xy+y^3)dx + (6y+2x^2+3xy^2)dy=0.
\end{eqnarray*}

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\item  %第4题
考虑一阶线性微分方程 $y' + 2xy = x$. 
\begin{enumerate}
\item  使用积分因子 $m(x)=\exp(x^2)$ 求解。
\item  将原方程写成如下形式，然后使用分离变量法求解，
\begin{eqnarray*}
\frac{dy}{dx} = x(1-2y). 
\end{eqnarray*}
\end{enumerate}

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\item  %第5题
求解微分方程 $\frac{dy}{dx} = \frac{y-x}{y+3x}$.

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\item  %第6题
考虑微分方程的初值问题 $\frac{dy}{dx} = xy+1, \, y(0)=1$.

\begin{enumerate}
\item  使用皮卡定理验证这个初值问题的解是存在且唯一的。
\item  写出皮卡序列的前三个函数。
\end{enumerate}

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\vspace{0.2cm}

\newpage
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\item  %第7题
求解微分方程 $\frac{dy}{dx} = y^2-1$. 并讨论在区域 $y>1$ 里的积分曲线的存在区间。

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\item  %第8题
考虑微分方程 $y=xp+p^2$, 其中 $p=\frac{dy}{dx}$. 
\begin{enumerate}
\item  使用微分法求解。
\item  使用 $p$-判别式求可能的奇解。
\item  按定义验证是不是奇解。
\end{enumerate}

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\end{enumerate}


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\end{document}

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